Jt. Rogers, DIOPHANTINE CONDITIONS IMPLY CRITICAL-POINTS ON THE BOUNDARIES OF SIEGEL DISKS OF POLYNOMIALS, Communications in Mathematical Physics, 195(1), 1998, pp. 175-193
Let f be a polynomial map of the Riemann sphere of degree at least two
, We prove that if f has: a Siegel disk G on which the rotation number
satisfies a diophantine condition, then either the boundary B of G co
ntains a critical point or B is a Lakes of Wada indecomposable continu
um with one of the lakes containing a critical point, Consequently, if
the boundary B of G has only 2 complementary domains, then B contains
a critical point. We also show, without any assumption on the rotatio
n number, that each proper nondegenerate subcontinuum of the boundary
B of G is tree-like, and any other bounded complementary domain of B i
s a preperiodic component of the grand orbit of G. Finally, we establi
sh some conditions under which B contains no periodic point.